In this paper a special conjugate Bayesian method, for reconstructing and estimating traffic flows, based on α-shifted-Gamma <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$ \Upgamma (\theta ,\,\lambda ) $$</EquationSource> </InlineEquation> models <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$ H(\alpha ,\,\theta ,\,\lambda ) $$</EquationSource> </InlineEquation> is given. If the numbers of users traveling through different routes are assumed to be independent <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$ H(\alpha ,\,\theta ,\,\lambda) $$</EquationSource> </InlineEquation> variables with common <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$ \lambda,$$</EquationSource> </InlineEquation> the link, origin–destination (OD) and node flows are also <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$ H(\alpha ,\,\theta ,\,\lambda ) $$</EquationSource> </InlineEquation> random variables. We assume that the main source of information is plate scanning, which permits us to identify, totally or partially, the vehicle route, OD and link flows by scanning their corresponding plate numbers at an adequately selected subset of links. The reconstruction of the sample flows can be done exactly or approximately, depending on the intensity of the plate scanning sampling procedure. To this end a generalized least squares technique is used together with the conservation laws. A Bayesian approach using special conjugate families is proposed that allows us to estimate different traffic flows, such as route, OD-pair, scanned link or counted link flows. A detailed description of how the prior assessment, the sampling, the posterior updating and the obtention of the Bayesian distribution is given. Finally, one example of application is used to illustrate the methods and procedures. Copyright Springer Science+Business Media New York 2013
